Abstract: Double contour integral formulas appear surprisingly often in expressions for correlations in various probabilistic models --- allowing for a swift asymptotic analysis via steepest descent analysis. Recent results on random tiling models demonstrate that such double contour formulas can also include integration on compact Riemann surfaces. As coalescence of saddle points characterizes transitions between different regions (like the smooth/rough boundary), we are naturally led to a notion of a discriminant on a Riemann surface, which I will introduce in this talk. It follows that such generalized discriminants  characterize arctic curves for various random tiling models, like the Aztec diamond and the hexagon. As a corollary we obtain degree formulas for arctic curves that depend only on the topology of the frozen, rough and smooth regions of the Aztec diamond (or hexagon), extending thereby the arctic circle theorem of Jockusch, Propp and Shor. This talk is based on the preprint arXiv:2410.17138.